No kids here, but some places to consider for dining that may work:

George Town Yacht Club - has a small pool and especially popular with families on the weekend.

Sunshine Grill at Sunshine Suites

Periwinkle at the Ritz - for family dinner & movie night

EATs - similar to an East Coast US style dinner.

Chicken! Chicken!

Ortanique at Camana Bay

Burger Shack.

You can find info and menus on most places on Grand Cayman here:

http://www.caymangoodtaste.com/